For the following data set, calculate the percentage of data points that fall within one standard deviation of the mean, and compare the result to the expected percentage of a normal distribution.
{50, 46, 54, 51, 29, 52, 48, 54, 47, 48}
A. 80%; This percentage is lower than expected in a normal distribution
B. 80%; This percentage is higher than expected in a normal distribution.
C. 90%; This percentage is lower than expected in a normal distribution.
D. 90%; This percentage is higher than expected in a normal distribution.
To calculate the percentage of data points that fall within one standard deviation of the mean, we need to find the standard deviation of the data set first.
The mean can be calculated as the sum of all the values divided by the total number of values:
mean = (50 + 46 + 54 + 51 + 29 + 52 + 48 + 54 + 47 + 48) / 10 = 499 / 10 = 49.9
Next, we calculate the differences between each value and the mean, squared them, and summed the result:
sum of (value - mean)^2 = (50-49.9)^2 + (46-49.9)^2 + (54-49.9)^2 + (51-49.9)^2 + (29-49.9)^2 + (52-49.9)^2 + (48-49.9)^2 + (54-49.9)^2 + (47-49.9)^2 + (48-49.9)^2
= 0.01 + 12.09 + 21.16 + 1.21 + 420.09 + 2.41 + 2.41 + 21.16 + 5.29 + 2.41
= 488.4
Then, we divide the sum by the total number of values minus 1 and take the square root to find the standard deviation:
standard deviation = sqrt(488.4 / (10-1))
= sqrt(488.4 / 9)
= sqrt(54.2667)
≈ 7.36
To find the number of data points within one standard deviation of the mean, we calculate the range from mean - standard deviation to mean + standard deviation:
lower bound = 49.9 - 7.36 = 42.54
upper bound = 49.9 + 7.36 = 57.26
The number of data points within this range is 6: 50, 46, 54, 51, 52, 48.
The percentage of data points within one standard deviation of the mean is (6 / 10) * 100% = 60%.
The expected percentage of data points within one standard deviation of the mean in a normal distribution is approximately 68%.
Therefore, the correct answer is:
C. 90%; This percentage is lower than expected in a normal distribution.